ANP-07: Horizon-Persistent Branch Extraction, Strong-Child Compactness, Causal Edge Closure, and the Renewal-Rate Alternative

Horizon-Persistent Branch Extraction, Strong-Child Compactness, Causal Edge Closure, and the Renewal-Rate Alternative

ANP-06 reduces strong-chain necessity to horizon-persistence obligations HP1, HP2, HP3, HP4. This round asks: what exactly prevents a horizon-persistent marked causal branch from being extracted? The answer now splits into two fundamentally different failure mechanisms: D_HTRANS (transmission degeneration) and D_HCOMP (compactness/edge-closure degeneration). Proves an abstract finite-child and compact-child horizon-persistent-child criterion — under a finite-threshold strong-child branch, HPC failure forces horizon-transmission collapse. Proves a horizon-cut dual-ledger theorem: if propagation inheritance is small on a contracting terminal window, the normalized fresh-source rate must diverge at least at the inverse-window-length rate. Under strong local compactness and a controlled kernel-inflated footprint, establishes a local causal-edge closure theorem for normalized-bounded-relative-frequency, bandwidth-limited C3_W edges — under a uniform transmission lower bound, a nontrivial edge limit holds. Critical-profile decomposition and the Type-I ancient-solution literature serve only as an external compactness calibration, and do not prove closure of the actual ancestry chain in the original solution.